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A recent measurement of the Lyman-limit mean free path at $z = 6$ suggests it may have been very short, motivating a better understanding of the role that ionizing photon sinks played in reionization. Accurately modeling the sinks in reionization simulations is challenging because of the large dynamic range required if $\sim 10^4-10^8 M_{\odot}$ gas structures contributed significant opacity. Thus, there is no consensus on how important the sinks were in shaping reionization's morphology. We address this question with a recently developed radiative transfer code that includes a dynamical sub-grid model for the sinks based on radiative hydrodynamics simulations. Compared to assuming a fully pressure-smoothed IGM, our dynamical treatment reduces ionized bubble sizes by $10-20\%$ under typical assumptions about reionization's sources. Near reionization's midpoint, the 21 cm power at $k \sim 0.1$ $h$Mpc$^{-1}$ is similarly reduced. These effects are more modest than the $30-60\%$ suppression resulting from the higher recombination rate if pressure smoothing is neglected entirely. Whether the sinks played a significant role in reionization's morphology depends on the nature of its sources. For example, if reionization was driven by bright ($M_{\rm UV} < -17$) galaxies, the sinks reduce the large-scale 21 cm power by at most $20\%$, even if pressure smoothing is neglected. Conveniently, when bright sources contribute significantly, the morphology in our dynamical treatment can be reproduced accurately with a uniform sub-grid clumping factor that yields the same ionizing photon budget. By contrast, if $M_{\rm UV} \sim -13$ galaxies drove reionization, the uniform clumping model can err by up to $40\%$.